45 46.] SPHERICAL BOUNDARY. 41 



pendent of the radius. It is therefore the same for any sphere, 

 concentric with the former one, which can be made to coincide 

 with it by gradual variation of the radius, without ever passing 

 out of the region occupied by the irrotationally moving liquid. 

 We may therefore suppose the sphere contracted to a point, and so 

 obtain a simple proof of the theorem, first given by Gauss in his 

 memoir* on the theory of Attractions, that the mean value of &amp;lt;f&amp;gt; 

 over any spherical surface throughout the interior of which (10) 

 is satisfied, is equal to its value at the centre. 



The theorem, proved in Art. 45, that &amp;lt;f&amp;gt; cannot be a maximum 

 or a minimum at a point in the interior of the fluid, is an obvious 

 consequence of the above. 



Again, let us suppose that the region occupied by the irrota- 

 tionally moving fluid is periphractic, -f- i.e. that it is limited in 

 ternally by one or more closed surfaces, and let us apply (11) to the 

 space included between one (or more) of these internal boundaries, 

 and a spherical surface completely enclosing it and lying wholly 

 in the fluid. If 4?rJ/ denote the total flux inwards across the 

 internal boundary of this space, we find, with the notation as 

 before, 



dr 



the surface integral extending over the sphere only. This may be 

 written 



1 d ff, 7 M 



- -y- I I (h citxr == n 

 whence 



That is, the mean value of &amp;lt;/&amp;gt; over any spherical surface drawn 

 under the above-mentioned conditions is equal to + C, where 



r is the radius, J/an absolute constant, and C a quantity which is 

 independent of the radius but may vary with the position of the 

 centre {. 



* Werke, t. 5, p. 199. A translation of the memoir is given in Taylor s Scientific 

 Memoirs, Vol. in. 



t See Maxwell, Electricity and Magnetism, Arts. 18, 22. 



It is understood, of course, that the spherical surfaces to which this statement 



